Find the angle 0 (in radians) between the vectors. sin + sin u = cos 4 v = cos 4 + sin 4
Find the angle 0 (in radians) between the vectors. sin + sin u = cos 4 v = cos 4 + sin 4
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Please see attached.
![**Finding the Angle Between Two Vectors**
Given the vectors:
\[ \mathbf{u} = \cos\left(\frac{\pi}{4}\right) \mathbf{i} + \sin\left(\frac{\pi}{4}\right) \mathbf{j} \]
\[ \mathbf{v} = \cos\left(\frac{5\pi}{4}\right) \mathbf{i} + \sin\left(\frac{5\pi}{4}\right) \mathbf{j} \]
We aim to find the angle \(\theta\) (in radians) between them.
**Step-by-Step Solution:**
1. **Identify Components of the Vectors:**
For vector \(\mathbf{u}\):
\[ u_i = \cos\left(\frac{\pi}{4}\right) \]
\[ u_j = \sin\left(\frac{\pi}{4}\right) \]
For vector \(\mathbf{v}\):
\[ v_i = \cos\left( \frac{5\pi}{4} \right) \]
\[ v_j = \sin\left( \frac{5\pi}{4} \right) \]
2. **Calculate the Dot Product of \(\mathbf{u}\) and \(\mathbf{v}\):**
\[ \mathbf{u} \cdot \mathbf{v} = \cos\left(\frac{\pi}{4}\right) \cos\left(\frac{5\pi}{4}\right) + \sin\left(\frac{\pi}{4}\right) \sin\left(\frac{5\pi}{4}\right) \]
3. **Calculate Magnitudes of \(\mathbf{u}\) and \(\mathbf{v}\):**
Magnitude of \(\mathbf{u}\):
\[ |\mathbf{u}| = \sqrt{ \cos^2\left(\frac{\pi}{4}\right) + \sin^2\left(\frac{\pi}{4}\right) } \]
Magnitude of \(\mathbf{v}\):
\[ |\mathbf{v}| = \sqrt{ \cos^2\left(\frac{5\pi}{4}\right) + \sin^2\left(\frac{5\pi}{4}\right) } \]
4. **Use](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9fe49cee-acc8-403b-8d12-20ec3514f583%2Fb3efb921-111b-469a-9edf-0fc04e0d8f88%2Fu567l1_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Angle Between Two Vectors**
Given the vectors:
\[ \mathbf{u} = \cos\left(\frac{\pi}{4}\right) \mathbf{i} + \sin\left(\frac{\pi}{4}\right) \mathbf{j} \]
\[ \mathbf{v} = \cos\left(\frac{5\pi}{4}\right) \mathbf{i} + \sin\left(\frac{5\pi}{4}\right) \mathbf{j} \]
We aim to find the angle \(\theta\) (in radians) between them.
**Step-by-Step Solution:**
1. **Identify Components of the Vectors:**
For vector \(\mathbf{u}\):
\[ u_i = \cos\left(\frac{\pi}{4}\right) \]
\[ u_j = \sin\left(\frac{\pi}{4}\right) \]
For vector \(\mathbf{v}\):
\[ v_i = \cos\left( \frac{5\pi}{4} \right) \]
\[ v_j = \sin\left( \frac{5\pi}{4} \right) \]
2. **Calculate the Dot Product of \(\mathbf{u}\) and \(\mathbf{v}\):**
\[ \mathbf{u} \cdot \mathbf{v} = \cos\left(\frac{\pi}{4}\right) \cos\left(\frac{5\pi}{4}\right) + \sin\left(\frac{\pi}{4}\right) \sin\left(\frac{5\pi}{4}\right) \]
3. **Calculate Magnitudes of \(\mathbf{u}\) and \(\mathbf{v}\):**
Magnitude of \(\mathbf{u}\):
\[ |\mathbf{u}| = \sqrt{ \cos^2\left(\frac{\pi}{4}\right) + \sin^2\left(\frac{\pi}{4}\right) } \]
Magnitude of \(\mathbf{v}\):
\[ |\mathbf{v}| = \sqrt{ \cos^2\left(\frac{5\pi}{4}\right) + \sin^2\left(\frac{5\pi}{4}\right) } \]
4. **Use
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